(square roots of the identity matrix) For how many 2x2 matrices A is it true that A^2=I ? Now answer the same question for n x n matrices where n>2

Chardonnay Felix 2021-03-08 Answered

(square roots of the identity matrix) For how many 2×2 matrices A is it true that A2=I ? Now answer the same question for n×x n matrices where n>2

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Expert Answer

au4gsf
Answered 2021-03-09 Author has 95 answers

Step 1
To find how many matricec A of order 2 exists such that A2=I
Here we have to find A2=A×A=I
Let us consider any arbitrary matrix of order 2. A=(abcd)
Then A2 is given by A2=(abcd)(abcd)=(a2+bcab+bdca+dcd2+bc)
Now from A2=I , we get A2=(a2+bcb(a+d)c(a+d)d2+bc)=(1001)=I
(1)a2+bc=1
(2)d2+bc=1
(3)b(a+d)=0
(3)c(a+d)=0
Step 2
Case 1:
If (a+d)0
From (3) we get : b=0
From (4) we get : c=0
From (1) we get : a2=1a=±1
From (2) we get : d2=1d=±1
Therefore we get the matrix of the form
{A=(a00d):a=±1,d=±1} Therefore there are only 4 matrices exist such that A2=I from case 1.
Step 3
Case 2:
If (a+d)=0
sub case a:
If a=0 , that implies d=0
Then from(1 ) and (2) , we get : bc=1c=b1=1b in real numbers.
Therefore we get the matrix of the form
{A=(0aa10):aF} Therefore there are infinitely many matrices exist if the field is the set of real numbers such that A2=I
Step 4
We can also consider further subcases to find the type of involuntary matrices, but as the question is asked to find how many, we have already got infinitely many 2×2 matrices of such type.
Similarly for any n>2 , we can find infinitely many matrices satisfying this condition on the field of real numbers.

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